WASHINGTON—Defense Department efforts to help Puerto Rico recover from Hurricane Maria will delay troop deployments to Afghanistan and affect deployments across the globe into the next year, Pentagon officials said Thursday, spurring questions about department-wide military readiness.

A planned increase in troops for the fight in Afghanistan will be delayed because of the response to the natural disaster, Lt. Gen. Kenneth McKenzie, the top staff officer on the Joint Chiefs of Staff told reporters. He said the delay doesn’t reflect unfavorably on the overall readiness of the U.S. military.

“Forces are flowing to Afghanistan,” Gen. McKenzie said, noting that the Pentagon has committed to adding nearly 4,000 troops to the approximately 11,000 U.S. forces currently in the country. “They have been slightly delayed by ongoing hurricane relief efforts,” he added.

Gen. McKenzie said U.S. forces have access only to a “finite number” of aircraft for transport, and their use in Puerto Rican relief efforts has caused a delay in forces to Afghanistan. He declined to provide further details of the delays, citing operational security issues.

Since the hurricane hit Puerto Rico, the Defense Department has deployed some 11,000 active duty and National Guard troops along with some 80 helicopters and more than 100 trucks to the island. A floating hospital, the USNS Comfort, also was sent.

Those troops represent a fraction of the hundreds of thousands of active duty service members and guardsmen at the Defense Department, which regularly cites its ability to respond across multiple fronts.

“The department’s big enough that it should be able to walk and chew gum at the same time,” said Phillip Carter, a former Army officer and now a senior fellow at the Center for a New American Security, a Washington defense think tank. He noted that the forces sent to Puerto Rico represent a fraction of available U.S. troop numbers.

With few further details coming from the Pentagon, Mr. Carter said the problem appeared to be a scarcity of certain assets, like airplanes and ships, rather than readiness of combat troops. Gen. McKenzie, asked about that question, said: “I’m not sure that you can naturally infer from that the force is unhealthy because we responded to a pretty significant natural disaster in Puerto Rico.”

Earlier this week, Defense Secretary Jim Mattis testified before a Senate committee that the hurricane response could interrupt the preparations of some units preparing to deploy overseas. “But when it comes to helping Americans, it’s all hands on deck,” he told senators, saying he was willing to accept the costs of hurricane response.

Gen. McKenzie said delays are to be expected after a major disaster response like Puerto Rico.

“There are going to be delays. I think the delays are relatively minor,” he said. “I’m not going to further elaborate on the nature of those delays.”

The Department of Defense has been criticized in recent months for readiness problems, with government auditors focusing on units like the Navy’s Seventh Fleet, based in Japan, which has been working sailors for long hours and has experienced a raft of accidents in recent months.